Which of the following worked equations demonstrates that you can divide polynomials by recognizing division as the inverse operation of multiplication?(1 point) Responses 8x2−4x+124x=(−4x)(8x2−4x+12) Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis negative 4 x right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis 8x2−4x+124x=(−14x)(8x2−4x+12) Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis negative Start Fraction 1 over 4 x End Fraction right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis 8x2−4x+124x=(14x)(8x2−4x+12) Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis Start Fraction 1 over 4 x End Fraction right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis 8x2−4x+124x=(4x)(8x2−4x+12)

1 answer

To demonstrate the concept of dividing polynomials using the inverse relationship of multiplication, we want to rearrange the division problem into a multiplication equation.

The expression we are working with is:

\[ \frac{8x^2 - 4x + 12}{4x} \]

We want to express it in the form \( A = (B)(C) \), where \( A \) is the original expression, \( B \) is the divisor, and \( C \) is the quotient.

None of the options you provided seem to match perfectly with a standard interpretation of the division as multiplication. However, the concept being illustrated is that you can express the division as a product where one factor is the reciprocal of the divisor.

So, if we consider:

\[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right) (8x^2 - 4x + 12) \]

This would correspond to a choice where:

\[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]

Given your options, the one that correctly demonstrates this relationship is:

Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis negative Start Fraction 1 over 4 x End Fraction right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis

Which can be simplified further if needed to express how dividing by \( 4x \) can be thought of as multiplying by its reciprocal.